368 Oresme, Nicole

ωωωωK

0,1,2,3,…,ω,ω+ 1, ω+ 2,…| = ω+ ω= ω×2

and

0,1,2,3,…,ω,ω+ 1,…,ω+ ω,ω+ ω+ 1,…| = ω+ ω+ ω

= ω×3

and, in continuing this process:

In this way, Cantor managed to develop an extraordi-

nary new type of arithmetic that extends the set of nat-

ural numbers to a system that includes transfinite

numbers. Since these quantities were derived from the

order property of natural numbers, he called these new

numbers ordinal numbers. Cantor developed clear and

precise rules for doing arithmetic with these numbers,

including adding, multiplying, and raising ordinal num-

bers to powers of each other. Cantor noted that the

ordinal number is the first ordinal number that

cannot be obtained from earlier ordinal numbers by a

finite number of additions, multiplications, and expo-

nentiations. He called this number ε0.

Following the approach taken in the study of

CAR

-

DINALITY

, Cantor also showed that the ordinal num-

bers can alternatively be constructed as follows. First

deem two sets, each of whose elements are ordered, to

be of the same ordinal number if there is a one-to-one

correspondence between the elements of the sets that

preserves the order of those elements. (For example, the

set of negative whole numbers and the set of positive

whole numbers have the same ordinal number via the

correspondence n↔– n.) Then ωis defined to be the

ordinal number of the set of natural numbers.

Oresme, Nicole (ca. 1323–1382) French Coordinate

geometry Born ca. 1323 in Allemagne, France, (the

exact birth date is not known) medieval scholar Nicole

Oresme is best remembered in mathematics for studies

of motion, and as the first scholar to depict a relation-

ship between two variables as a

GRAPH OF A FUNCTION

.

Oresme studied A

RISTOTLE

’s theory of motion, kine-

matics, at the University of Paris under the guidance of

philosopher and logician Jean Buridan. He received an

arts degree in the early 1340s and later went on to

obtain a master’s degree in theology at the same institu-

tion in 1355. Although Oresme followed a career path

dedicated to work in a Catholic order (he was later

appointed bishop of Lisieux), Oresme continued to pur-

sue an active interest in the work of Aristotle throughout

his life and published important works that influenced

scholarly thinking on this subject. Oresme found philo-

sophical difficulties with Aristotle’s proposed definitions

of time and space, which themselves were dependent on

the notion of movement, and proposed alternative defi-

nitions independent of this concept.

Perhaps Oresme’s most important work is De con-

figurationibus qualitatum et motuum (The geometry of

qualities and motion), in which he describes, for the first

time, a general procedure for representing relationships

between variables pictorially. Moreover, he realized that

the area under the graph of a uniformly varying quantity

represents the total change of the quantity. Oresme had

consequently invented a type of coordinate geometry

and made first steps toward work in integral

CALCULUS

.

(In this same work, Oresme also proved that the dis-

tance traveled over a fixed time by an object moving

with constant acceleration is the same as for an object

moving at uniform velocity equal to the speed of the first

object at the midpoint of the time period. This is a

remarkable achievement given that the tools and tech-

niques of calculus were not available to him at the time.)

Oresme also studied infinite

SERIES

, often using

ingenious graphical tricks to establish results. The stan-

dard proof of the divergence of the

HARMONIC SERIES

(by grouping the terms of the series and comparing

with sums of one-half) is due to him.

His studies on the motions of the planets also led

him to study proportions and

RATIONAL NUMBERS

.

Oresme was the first scholar in the history of mathe-

matics to consider, and work with, fractional expo-

nents. Like the scholars of antiquity Oresme sought for

harmony in the universe. He proposed, for instance,

that the ratio of the periods of any two heavenly bodies

will always have a rational value. (This is not the case.)

Oresme died in Lisieux, France, in 1382. The exact

death date is not known.